3.543 \(\int \frac{x^5}{\sqrt{9-4 x^2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{320} \left (9-4 x^2\right )^{5/2}+\frac{3}{32} \left (9-4 x^2\right )^{3/2}-\frac{81}{64} \sqrt{9-4 x^2} \]

[Out]

(-81*Sqrt[9 - 4*x^2])/64 + (3*(9 - 4*x^2)^(3/2))/32 - (9 - 4*x^2)^(5/2)/320

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Rubi [A]  time = 0.0188897, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{1}{320} \left (9-4 x^2\right )^{5/2}+\frac{3}{32} \left (9-4 x^2\right )^{3/2}-\frac{81}{64} \sqrt{9-4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[9 - 4*x^2],x]

[Out]

(-81*Sqrt[9 - 4*x^2])/64 + (3*(9 - 4*x^2)^(3/2))/32 - (9 - 4*x^2)^(5/2)/320

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{9-4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16 \sqrt{9-4 x}}-\frac{9}{8} \sqrt{9-4 x}+\frac{1}{16} (9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{81}{64} \sqrt{9-4 x^2}+\frac{3}{32} \left (9-4 x^2\right )^{3/2}-\frac{1}{320} \left (9-4 x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0082703, size = 27, normalized size = 0.59 \[ -\frac{1}{40} \sqrt{9-4 x^2} \left (2 x^4+6 x^2+27\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[9 - 4*x^2],x]

[Out]

-(Sqrt[9 - 4*x^2]*(27 + 6*x^2 + 2*x^4))/40

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Maple [A]  time = 0.003, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 2\,{x}^{4}+6\,{x}^{2}+27 \right ) }{40}{\frac{1}{\sqrt{-4\,{x}^{2}+9}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-4*x^2+9)^(1/2),x)

[Out]

1/40*(-3+2*x)*(3+2*x)*(2*x^4+6*x^2+27)/(-4*x^2+9)^(1/2)

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Maxima [A]  time = 1.80632, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{20} \, \sqrt{-4 \, x^{2} + 9} x^{4} - \frac{3}{20} \, \sqrt{-4 \, x^{2} + 9} x^{2} - \frac{27}{40} \, \sqrt{-4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/20*sqrt(-4*x^2 + 9)*x^4 - 3/20*sqrt(-4*x^2 + 9)*x^2 - 27/40*sqrt(-4*x^2 + 9)

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Fricas [A]  time = 1.27043, size = 61, normalized size = 1.33 \begin{align*} -\frac{1}{40} \,{\left (2 \, x^{4} + 6 \, x^{2} + 27\right )} \sqrt{-4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

-1/40*(2*x^4 + 6*x^2 + 27)*sqrt(-4*x^2 + 9)

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Sympy [A]  time = 1.15974, size = 46, normalized size = 1. \begin{align*} - \frac{x^{4} \sqrt{9 - 4 x^{2}}}{20} - \frac{3 x^{2} \sqrt{9 - 4 x^{2}}}{20} - \frac{27 \sqrt{9 - 4 x^{2}}}{40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-4*x**2+9)**(1/2),x)

[Out]

-x**4*sqrt(9 - 4*x**2)/20 - 3*x**2*sqrt(9 - 4*x**2)/20 - 27*sqrt(9 - 4*x**2)/40

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Giac [A]  time = 2.93596, size = 58, normalized size = 1.26 \begin{align*} -\frac{1}{320} \,{\left (4 \, x^{2} - 9\right )}^{2} \sqrt{-4 \, x^{2} + 9} + \frac{3}{32} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} - \frac{81}{64} \, \sqrt{-4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

-1/320*(4*x^2 - 9)^2*sqrt(-4*x^2 + 9) + 3/32*(-4*x^2 + 9)^(3/2) - 81/64*sqrt(-4*x^2 + 9)